Infinite products of automorphic forms

二人主要从事自同构形式的数论，特别是模函数的无穷积和各种自同构形式的提升理论的研究。对于q展开式的实系数模函数，我们知道系数的符号序列通常是周期性的。Thompson系列的例子由McKay- Strauss观察到，一些特殊的例子包括1/j由第一作者和Kaneko-Ninomiya处理。在无限积型的许多情况下，McKay-Strauss型的符号模式可以用Hardy-Ramanajan的圆法来解释。事实上，系数的符号与Kloosterman和的符号是一致的所以它们是周期性的。第一作者处理了许多其他一般无限积的多符号模式，特别是发现有时会出现相应的系数都周期性为零的情况，这涉及到某一般Kloosterman和的消失问题。已知存在从GL(2)-自同构形式到GL(3)-形式的提升。事实上，Gelbart & Jaquet已经用l函数的泛函方程方法给出了第一个构造。第二作者成功地用对应方法重建了提升，其中他使用了新的艾森斯滕级数的积分表达式。这种新方法的显著成功将为许多其他举重运动提供新的启示。在另一个方向上，第二作者还对一些3次Sigel模形式的傅里叶系数进行了大量的计算，这有力地支持了miyaaki举升的一个猜想。

The research project has been pursued by two authors mainly on number theory of automorphic forms, especially on the infinite products of modular functions and on lifting theory of various automorphic forms.1. Concerning the modular function with real coefficients of the q-expansion, it is known that the sequences of signs of the coefficients are very often periodic. The cases of Thompson series are observed by McKay- Strauss, and some special cases including 1/j are treated by the first author and Kaneko-Ninomiya.On many cases of the infinite products type or so, the sign patterns of McKay-Strauss' type can be explained by the circle method of Hardy-Ramanajan. In fact it can be shown the signs of coefficients coincide with the signs of Kloosterman sums and so they are periodic.The first author treated the more sign patterns of many other infinite products of general type, and in particular it was found that it sometimes happen the corresponding coefficients are all zero periodically, which relates to the vanishing problem of a certain general Kloosterman sum.2. It is known there exists a lifting from GL(2)-automorphic forms to GL(3)-forms. In fact, Gelbart & Jaquet already gave the first construction by using the functional equations method of L-functions.The second author has succeeded in reconstruction of the lifting by theta correspondence method, where he uses new integral expression of Eisensten series. This remarkable success of quite new method will shed new light on many other liftings.On another direction, the second author also executed a big calculation of Fourier coefficients of some Sigel modular forms of degree 3, which supports strongly a conjecture of Miyawaki lifting.